Problem 62 - Elementary Algebra

Question

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: The relationship of the number of weeks a box of garbage bags is used, \(x\), and the number of bags left in the box, \(y\), is an inverse variation. When \(x\) is 8 weeks, \(y\) is 168 bags. Find the constant of proportionality, \(k\). Incorrect Answer: \(k=\frac{y}{x}\) $$ k=\frac{168 \text { bags }}{8 \text { weeks }} $$

Step-by-Step Solution

Answer

The mistake is using the direct variation formula. Correct answer: $$k = 1344$$

Step 1 - Identify the Mistake

In an inverse variation, the relationship between two variables is given by the equation $$ xy = k $$ where $$ k $$ is the constant of proportionality. The mistake in the solution is the use of the formula $$ k = \frac{y}{x} $$ which applies to direct variation, not inverse variation.

Step 2 - Use the Correct Formula

For inverse variation, we use the formula $$ xy = k $$ which means that the product of the variables $$ x $$ and $$ y $$ is equal to the constant $$ k $$.

Step 3 - Substitute the Given Values

The problem states that $$ x = 8 \text { weeks } $$ and $$ y = 168 \text { bags } $$. Substitute these values into the formula to find $$ k $$: $$ k = xy = 8 \text { weeks } \times 168 \text { bags } $$

Step 4 - Calculate the Constant of Proportionality

Perform the multiplication to find $$ k $$: $$ k = 8 \times 168 = 1344 \text { (weeks * bags) } $$

Key Concepts

inverse variation

Inverse variation describes a relationship between two variables where their product is constant. This means as one variable increases, the other decreases proportionally. The mathematical formula is given by:

where $$ xy = k $$ Here, k is the constant of proportionality. When working with inverse variation, remember:

• The product of the variables stays constant.
• As one variable increases, the other decreases.


• The formula differs from direct variation.
  Direct variation uses the formula < right> $$ y = kx $$

Remember to always identify whether a relationship is direct or inverse before choosing the correct formula.

constant of proportionality

The constant of proportionality, denoted as $$ k $$, is a crucial value in both direct and inverse proportional relationships. In the context of inverse variation, it signifies that the product of the variables does not change. For instance, if you have an inverse variation problem and you know the values of the two variables, you can find $$ k $$ by multiplying these values together.

Here’s how we cracked this in our problem:

• The given values were $$ x = 8 \text { weeks } $$ and $$ y = 168 \text { bags } $$.
• We used the correct formula for inverse variation: $$ xy = k $$
• Substituted the values: $$ k = 8 \times 168 $$
• Calculated to find $$ k = 1344 \text { (weeks * bags) } $$

Understanding $$ k $$ helps you to quickly find relationships between variables and solve inverse variation problems with confidence.

algebraic equations

Algebraic equations are mathematical statements that use numbers and variables. They set two expressions equal to each other. Inversely varying relationships involve algebraic equations like $$ xy = k $$. Here’s a recap of key points:

• Identify the type of relation (inverse or direct).


• For inverse variation, multiply the variables to equal the constant.


• Isolate the unknown variable if needed to solve for it.

Example from our problem:

To find the constant of proportionality, we knew that:

• $$ k = xy $$


• The given values were $$ x = 8 $$ and $$ y = 168 $$.


• By plugging in these values, we found $$ k = 1344 $$ through straightforward multiplication.

By practicing these steps, solving algebraic equations becomes much easier!

error identification

Identifying errors in mathematical solutions is crucial to mastering algebra. Inverse variation problems often involve a specific common mistake: confusing inverse and direct variation formulas. In our given problem, the error was using
$$ \frac{y}{x} $$ instead of the inverse variation formula $$ xy = k $$.

To avoid this, follow these tips:

• Clearly understand the type of variation described (inverse or direct).


• Use the correct formula from the start. For inverse: $$ xy = k $$.For direct: $$ y = kx $$


• Double-check your formula before computing.


• Review the problem context if your result seems off.

In this case, identifying and correcting the mistake was simple:

• We recognized the given formula $$ k = \frac{y}{x} $$ was wrong for inverse variation.


• Used the correct formula: $$ xy = k $$


• Calculated accurately to find $$ k = 8 \times 168 = 1344 $$

Consistently practice error identification to enhance your problem-solving skills!